How to easily draw an octagon with 8 equal sides (equilateral octagon) without doing any calculations other than measuring the size of the square that will be used to draw the octagon. An explanation of how this works is also included so the student learning geometry will know the steps in the process of how this is done.

Draw a square the same size as the octagon that will be drawn (in this example the square has 5 inch sides). Draw two lines from corner to corner making an "X".

Using another piece of paper, place one edge on the intersection of the "X" and put a mark at one corner of the square.

**A ruler can also be used for this step, just note the measurement between the "X" and corner.

A compass can also be used for this step. Set the point of the compass on one of the corners of the square and open it to the "X".

Turn the piece of paper and with the mark at the corner of the square, put a mark on the square at the edge of the piece of paper. Continue with both sides of all corners until there are eight (8) total marks on the square.

**If using a compass, with the point on each corner of the square, make two marks on each adjacent side of the square for eight total marks.

**If using a ruler, measure from each corner the same distance as in Step 2.

Draw a line between the two mark nearest each corner and erase the corners of the square and the "X" to complete the equilateral octagon.

HOW IT WORKS: Using Pythagoreans Theorem, which is A²+B²=C², calculate the length of the hypotenuse, or "C" in the picture. The length of one side of the square is 5 inches, so 1/2 this length is 2-1/2". Since all sides of the square are equal, "A" and "B" are both 2-1/2". This is the equation:

(2.5)²+(2.5)²= C²

6.25 + 6.25 = 12.5. The square root of 12.5 is 3.535 so "C" = 3.535.

In Step 4 a mark was placed 3.535" from each corner of the square which is a distance of 1.4645" ("AA" in the picture) from the opposite corner.

5 - C = AA. So "AA" = 1.4645.

Since each mark is 1.4645" from each corner of the square. Subtract two of these measurements from the side of the square to obtain the length of the side of the octagon (CC):

5 - (1.4645 * 2) = CC.

5 - 2.929 = CC

CC = 2.071.

Use Pythagoreans Theorem to double check the length of the hypotenuse of the triangle "AA-BB-CC" in the picture (AA and BB are equal, or 1.4645):

AA² + BB² = CC²

1.4645² + 1.4645² = CC²

2.145 + 2.145 = 4.289².

The square root of 4.289 is 2.071, which is equal to the step above, confirming this is an equilateral octagon.